Loveth Kirika was driving her 2007 Chevrolet HHR toward Houston in April when it suddenly stalled on Interstate 45. Another car rear-ended her, knocking out the 39-year-old woman with a concussion.

In May, her husband, Lenu, got a telephone call from Mark Byrd, an investigator of product-liability cases at General Motors Co. He wanted to know where the car’s black box was. Mr. Kirika replied that he hadn’t seen the car since it was…